On the Efficiency of Fair and Truthful Trade Mechanisms

TL;DR

The paper studies how imposing KS-fairness constraints on truthful bilateral-trade mechanisms affects social efficiency in Bayesian settings. It introduces KS-fairness as ex ante equalization of each trader’s fraction of their respective ideal utilities, and shows a general $1/2$-approximation to the Second-Best GFT benchmark is achievable, with tightness results. In zero-value-seller scenarios, and under Regular or MHR distributions for the buyer, it derives substantially higher GFT fractions via simple mechanisms such as Fixed Price and the $\lambda$-Biased Random Offer Mechanism, supported by revenue-curve and hazard-rate analyses and numerical optimization. The work also links KS-fairness to Nash social welfare, establishing a $1/2$-approximation bound with corresponding tight lower bounds, and discusses implications for market regulation and future generalizations to broader bargaining settings and multi-agent markets.

Abstract

We consider the impact of fairness requirements on the social efficiency of truthful mechanisms for trade, focusing on Bayesian bilateral-trade settings. Unlike the full information case in which all gains-from-trade can be realized and equally split between the two parties, in the private information setting, equitability has devastating welfare implications (even if only required to hold ex-ante). We thus search for an alternative fairness notion and suggest requiring the mechanism to be KS-fair: it must ex-ante equalize the fraction of the ideal utilities of the two traders. We show that there is always a KS-fair (simple) truthful mechanism with expected gains-from-trade that are half the optimum, but always ensuring any better fraction is impossible (even when the seller value is zero). We then restrict our attention to trade settings with a zero-value seller and a buyer with value distribution that is Regular or MHR, proving that much better fractions can be obtained under these conditions.

Figures (6)

• Figure 1: Graphical illustration of \ref{['example:all fair mech:irregular']}. In \ref{['fig:all fair mech:irregular:revenue curve']}, the black curve is the revenue curve of the buyer. In \ref{['fig:all fair mech:irregular:ex ante utility pair']}, the shaded region corresponds to all pairs of buyer and seller's ex ante utilities $(U,\Pi)$ that are achievable by some BIC, IIR, ex ante WBB mechanism. The two black points represent the traders' ex ante utility pairs $(U^*,0)$ and $(0,\Pi^*)$, induced by the Buyer Offer Mechanism and Seller Offer Mechanism, respectively. The red dashed line represents utility pairs that are KS-fair (i.e., ones with $U/U^* = \Pi/\Pi^*$). The red square corresponds to a truthful KS-fair mechanism whose GFT is at least $\frac{1}{2}$ fraction of the Second-Best Benchmark$\texttt{OPT}_{\textsc{SB}}$.
• Figure 2: Graphical illustration of \ref{['example:BROM:regular']} when $K = 25$. The x-axis is quantile $q$. The black solid line is the revenue curve of the buyer. Consider Fixed Price Mechanism$\mathcal{M}$ with trading price $p = v(q)$ for every quantile $q$. The black solid (resp. dashed) curve also represents ${\Pi(\mathcal{M})}/{\Pi^*}$ (resp. ${U(\mathcal{M})}/{U^*}$) for the seller (resp. buyer). The red curve is the GFT approximation ratio ${\texttt{GFT}_{}\!\left[{\mathcal{M}}\right]}/{\texttt{OPT}_{\textsc{SB}}}$. A KS-fair Fixed Price Mechanism is achieved at trading price $p_f = v(q_f)$ with GFT approximation ratio of $0.877$ (when $K = 25$). Finally, the blue curve is used in the proof of the negative result in \ref{['lem:GFT UB:regular buyer']}.
• Figure 3: Graphical illustration of the analysis for \ref{['lem:GFT program:regular buyer']}. The black curve is the concave revenue curve $R$ of the buyer. The red and blue revenue curves $R_1, R_2$ (defined in the analysis) sandwich the original revenue curve $R$.
• Figure 4: Graphical illustration of the analysis for \ref{['lem:GFT program:mhr buyer']}. The black curve is the convex cumulative hazard rate function $\Phi$ of the buyer. The red and blue cumulative hazard rate functions $\Phi_1, \Phi_2$ (defined in the analysis) sandwich the original cumulative hazard rate function $\Phi$. The black dashed line is $\ln(v)$, which touches the cumulative hazard rate function $\Phi$ at monopoly reserve $r_m$.
• Figure 5: Graphical illustration of \ref{['example:all fair:mhr buyer']}. The x-axis is value $v$. Consider Fixed Price Mechanism$\mathcal{M}$ with trading price $p$ for every $p\in[0,e]$. The black solid (resp. dashed) curve also represents ${\Pi(\mathcal{M})}/{\Pi^*}$ (resp. ${U(\mathcal{M})}/{U^*}$) for the seller (resp. buyer). The red curve is the GFT approximation ratio ${\texttt{GFT}_{}\!\left[{\mathcal{M}}\right]}/{\texttt{OPT}_{\textsc{SB}}}$. A KS-fair Fixed Price Mechanism is achieved at trading price $p_f \approx 0.80$ with GFT approximation ratio of $0.944$. Finally, the blue curve is used in the proof of the negative result in \ref{['lem:GFT UB:mhr buyer']}.

Theorems & Definitions (72)

• Definition 2.1: Regularity and MHR for the buyer
• Definition 2.2: Revenue curve
• Lemma 2.1: BR-89
• Proposition 2.2: mye-81HR-08
• Definition 3.1: KS-fairness
• Lemma 3.0
• proof
• Definition 4.1: $\lambda$-ROM
• Theorem 4.1
• Theorem 4.2: Black-box reduction